geevx(3)

NAME
geevx - geevx: eig, expert

SYNOPSIS

  Functions
subroutine cgeevx (balanc, jobvl, jobvr, sense, n, a, lda, w, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, rwork, info)
CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine dgeevx (balanc, jobvl, jobvr, sense, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, iwork, info)
DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sgeevx (balanc, jobvl, jobvr, sense, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, iwork, info)
SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine zgeevx (balanc, jobvl, jobvr, sense, n, a, lda, w, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, rwork, info)
ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Detailed Description

Function Documentation

  subroutine cgeevx (character balanc, character jobvl, character jobvr, character sense, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) w, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, real, dimension( * ) scale, real abnrm, real, dimension( * ) rconde, real, dimension( * ) rcondv, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)

CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

    !>
    !> CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
    !> eigenvalues and, optionally, the left and/or right eigenvectors.
    !>
    !> Optionally also, it computes a balancing transformation to improve
    !> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
    !> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
    !> (RCONDE), and reciprocal condition numbers for the right
    !> eigenvectors (RCONDV).
    !>
    !> The right eigenvector v(j) of A satisfies
    !>                  A * v(j) = lambda(j) * v(j)
    !> where lambda(j) is its eigenvalue.
    !> The left eigenvector u(j) of A satisfies
    !>               u(j)**H * A = lambda(j) * u(j)**H
    !> where u(j)**H denotes the conjugate transpose of u(j).
    !>
    !> The computed eigenvectors are normalized to have Euclidean norm
    !> equal to 1 and largest component real.
    !>
    !> Balancing a matrix means permuting the rows and columns to make it
    !> more nearly upper triangular, and applying a diagonal similarity
    !> transformation D * A * D**(-1), where D is a diagonal matrix, to
    !> make its rows and columns closer in norm and the condition numbers
    !> of its eigenvalues and eigenvectors smaller.  The computed
    !> reciprocal condition numbers correspond to the balanced matrix.
    !> Permuting rows and columns will not change the condition numbers
    !> (in exact arithmetic) but diagonal scaling will.  For further
    !> explanation of balancing, see section 4.10.2 of the LAPACK
    !> Users' Guide.
    !>

Parameters
    BALANC

    !>          BALANC is CHARACTER*1
    !>          Indicates how the input matrix should be diagonally scaled
    !>          and/or permuted to improve the conditioning of its
    !>          eigenvalues.
    !>          = 'N': Do not diagonally scale or permute;
    !>          = 'P': Perform permutations to make the matrix more nearly
    !>                 upper triangular. Do not diagonally scale;
    !>          = 'S': Diagonally scale the matrix, ie. replace A by
    !>                 D*A*D**(-1), where D is a diagonal matrix chosen
    !>                 to make the rows and columns of A more equal in
    !>                 norm. Do not permute;
    !>          = 'B': Both diagonally scale and permute A.
    !>
    !>          Computed reciprocal condition numbers will be for the matrix
    !>          after balancing and/or permuting. Permuting does not change
    !>          condition numbers (in exact arithmetic), but balancing does.
    !>

    JOBVL

    !>          JOBVL is CHARACTER*1
    !>          = 'N': left eigenvectors of A are not computed;
    !>          = 'V': left eigenvectors of A are computed.
    !>          If SENSE = 'E' or 'B', JOBVL must = 'V'.
    !>

    JOBVR

    !>          JOBVR is CHARACTER*1
    !>          = 'N': right eigenvectors of A are not computed;
    !>          = 'V': right eigenvectors of A are computed.
    !>          If SENSE = 'E' or 'B', JOBVR must = 'V'.
    !>

    SENSE

    !>          SENSE is CHARACTER*1
    !>          Determines which reciprocal condition numbers are computed.
    !>          = 'N': None are computed;
    !>          = 'E': Computed for eigenvalues only;
    !>          = 'V': Computed for right eigenvectors only;
    !>          = 'B': Computed for eigenvalues and right eigenvectors.
    !>
    !>          If SENSE = 'E' or 'B', both left and right eigenvectors
    !>          must also be computed (JOBVL = 'V' and JOBVR = 'V').
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A. N >= 0.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>          On entry, the N-by-N matrix A.
    !>          On exit, A has been overwritten.  If JOBVL = 'V' or
    !>          JOBVR = 'V', A contains the Schur form of the balanced
    !>          version of the matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    W

    !>          W is COMPLEX array, dimension (N)
    !>          W contains the computed eigenvalues.
    !>

    VL

    !>          VL is COMPLEX array, dimension (LDVL,N)
    !>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
    !>          after another in the columns of VL, in the same order
    !>          as their eigenvalues.
    !>          If JOBVL = 'N', VL is not referenced.
    !>          u(j) = VL(:,j), the j-th column of VL.
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of the array VL.  LDVL >= 1; if
    !>          JOBVL = 'V', LDVL >= N.
    !>

    VR

    !>          VR is COMPLEX array, dimension (LDVR,N)
    !>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
    !>          after another in the columns of VR, in the same order
    !>          as their eigenvalues.
    !>          If JOBVR = 'N', VR is not referenced.
    !>          v(j) = VR(:,j), the j-th column of VR.
    !>

    LDVR

    !>          LDVR is INTEGER
    !>          The leading dimension of the array VR.  LDVR >= 1; if
    !>          JOBVR = 'V', LDVR >= N.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>          ILO and IHI are integer values determined when A was
    !>          balanced.  The balanced A(i,j) = 0 if I > J and
    !>          J = 1,...,ILO-1 or I = IHI+1,...,N.
    !>

    SCALE

    !>          SCALE is REAL array, dimension (N)
    !>          Details of the permutations and scaling factors applied
    !>          when balancing A.  If P(j) is the index of the row and column
    !>          interchanged with row and column j, and D(j) is the scaling
    !>          factor applied to row and column j, then
    !>          SCALE(J) = P(J),    for J = 1,...,ILO-1
    !>                   = D(J),    for J = ILO,...,IHI
    !>                   = P(J)     for J = IHI+1,...,N.
    !>          The order in which the interchanges are made is N to IHI+1,
    !>          then 1 to ILO-1.
    !>

    ABNRM

    !>          ABNRM is REAL
    !>          The one-norm of the balanced matrix (the maximum
    !>          of the sum of absolute values of elements of any column).
    !>

    RCONDE

    !>          RCONDE is REAL array, dimension (N)
    !>          RCONDE(j) is the reciprocal condition number of the j-th
    !>          eigenvalue.
    !>

    RCONDV

    !>          RCONDV is REAL array, dimension (N)
    !>          RCONDV(j) is the reciprocal condition number of the j-th
    !>          right eigenvector.
    !>

    WORK

    !>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.  If SENSE = 'N' or 'E',
    !>          LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
    !>          LWORK >= N*N+2*N.
    !>          For good performance, LWORK must generally be larger.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    RWORK

    !>          RWORK is REAL array, dimension (2*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if INFO = i, the QR algorithm failed to compute all the
    !>                eigenvalues, and no eigenvectors or condition numbers
    !>                have been computed; elements 1:ILO-1 and i+1:N of W
    !>                contain eigenvalues which have converged.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 285 of file cgeevx.f.

  subroutine dgeevx (character balanc, character jobvl, character jobvr, character sense, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, double precision, dimension( * ) scale, double precision abnrm, double precision, dimension( * ) rconde, double precision, dimension( * ) rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

    !>
    !> DGEEVX computes for an N-by-N real nonsymmetric matrix A, the
    !> eigenvalues and, optionally, the left and/or right eigenvectors.
    !>
    !> Optionally also, it computes a balancing transformation to improve
    !> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
    !> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
    !> (RCONDE), and reciprocal condition numbers for the right
    !> eigenvectors (RCONDV).
    !>
    !> The right eigenvector v(j) of A satisfies
    !>                  A * v(j) = lambda(j) * v(j)
    !> where lambda(j) is its eigenvalue.
    !> The left eigenvector u(j) of A satisfies
    !>               u(j)**H * A = lambda(j) * u(j)**H
    !> where u(j)**H denotes the conjugate-transpose of u(j).
    !>
    !> The computed eigenvectors are normalized to have Euclidean norm
    !> equal to 1 and largest component real.
    !>
    !> Balancing a matrix means permuting the rows and columns to make it
    !> more nearly upper triangular, and applying a diagonal similarity
    !> transformation D * A * D**(-1), where D is a diagonal matrix, to
    !> make its rows and columns closer in norm and the condition numbers
    !> of its eigenvalues and eigenvectors smaller.  The computed
    !> reciprocal condition numbers correspond to the balanced matrix.
    !> Permuting rows and columns will not change the condition numbers
    !> (in exact arithmetic) but diagonal scaling will.  For further
    !> explanation of balancing, see section 4.10.2 of the LAPACK
    !> Users' Guide.
    !>

Parameters
    BALANC

    !>          BALANC is CHARACTER*1
    !>          Indicates how the input matrix should be diagonally scaled
    !>          and/or permuted to improve the conditioning of its
    !>          eigenvalues.
    !>          = 'N': Do not diagonally scale or permute;
    !>          = 'P': Perform permutations to make the matrix more nearly
    !>                 upper triangular. Do not diagonally scale;
    !>          = 'S': Diagonally scale the matrix, i.e. replace A by
    !>                 D*A*D**(-1), where D is a diagonal matrix chosen
    !>                 to make the rows and columns of A more equal in
    !>                 norm. Do not permute;
    !>          = 'B': Both diagonally scale and permute A.
    !>
    !>          Computed reciprocal condition numbers will be for the matrix
    !>          after balancing and/or permuting. Permuting does not change
    !>          condition numbers (in exact arithmetic), but balancing does.
    !>

    JOBVL

    !>          JOBVL is CHARACTER*1
    !>          = 'N': left eigenvectors of A are not computed;
    !>          = 'V': left eigenvectors of A are computed.
    !>          If SENSE = 'E' or 'B', JOBVL must = 'V'.
    !>

    JOBVR

    !>          JOBVR is CHARACTER*1
    !>          = 'N': right eigenvectors of A are not computed;
    !>          = 'V': right eigenvectors of A are computed.
    !>          If SENSE = 'E' or 'B', JOBVR must = 'V'.
    !>

    SENSE

    !>          SENSE is CHARACTER*1
    !>          Determines which reciprocal condition numbers are computed.
    !>          = 'N': None are computed;
    !>          = 'E': Computed for eigenvalues only;
    !>          = 'V': Computed for right eigenvectors only;
    !>          = 'B': Computed for eigenvalues and right eigenvectors.
    !>
    !>          If SENSE = 'E' or 'B', both left and right eigenvectors
    !>          must also be computed (JOBVL = 'V' and JOBVR = 'V').
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A. N >= 0.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the N-by-N matrix A.
    !>          On exit, A has been overwritten.  If JOBVL = 'V' or
    !>          JOBVR = 'V', A contains the real Schur form of the balanced
    !>          version of the input matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    WR

    !>          WR is DOUBLE PRECISION array, dimension (N)
    !>

    WI

    !>          WI is DOUBLE PRECISION array, dimension (N)
    !>          WR and WI contain the real and imaginary parts,
    !>          respectively, of the computed eigenvalues.  Complex
    !>          conjugate pairs of eigenvalues will appear consecutively
    !>          with the eigenvalue having the positive imaginary part
    !>          first.
    !>

    VL

    !>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
    !>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
    !>          after another in the columns of VL, in the same order
    !>          as their eigenvalues.
    !>          If JOBVL = 'N', VL is not referenced.
    !>          If the j-th eigenvalue is real, then u(j) = VL(:,j),
    !>          the j-th column of VL.
    !>          If the j-th and (j+1)-st eigenvalues form a complex
    !>          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
    !>          u(j+1) = VL(:,j) - i*VL(:,j+1).
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of the array VL.  LDVL >= 1; if
    !>          JOBVL = 'V', LDVL >= N.
    !>

    VR

    !>          VR is DOUBLE PRECISION array, dimension (LDVR,N)
    !>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
    !>          after another in the columns of VR, in the same order
    !>          as their eigenvalues.
    !>          If JOBVR = 'N', VR is not referenced.
    !>          If the j-th eigenvalue is real, then v(j) = VR(:,j),
    !>          the j-th column of VR.
    !>          If the j-th and (j+1)-st eigenvalues form a complex
    !>          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
    !>          v(j+1) = VR(:,j) - i*VR(:,j+1).
    !>

    LDVR

    !>          LDVR is INTEGER
    !>          The leading dimension of the array VR.  LDVR >= 1, and if
    !>          JOBVR = 'V', LDVR >= N.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>          ILO and IHI are integer values determined when A was
    !>          balanced.  The balanced A(i,j) = 0 if I > J and
    !>          J = 1,...,ILO-1 or I = IHI+1,...,N.
    !>

    SCALE

    !>          SCALE is DOUBLE PRECISION array, dimension (N)
    !>          Details of the permutations and scaling factors applied
    !>          when balancing A.  If P(j) is the index of the row and column
    !>          interchanged with row and column j, and D(j) is the scaling
    !>          factor applied to row and column j, then
    !>          SCALE(J) = P(J),    for J = 1,...,ILO-1
    !>                   = D(J),    for J = ILO,...,IHI
    !>                   = P(J)     for J = IHI+1,...,N.
    !>          The order in which the interchanges are made is N to IHI+1,
    !>          then 1 to ILO-1.
    !>

    ABNRM

    !>          ABNRM is DOUBLE PRECISION
    !>          The one-norm of the balanced matrix (the maximum
    !>          of the sum of absolute values of elements of any column).
    !>

    RCONDE

    !>          RCONDE is DOUBLE PRECISION array, dimension (N)
    !>          RCONDE(j) is the reciprocal condition number of the j-th
    !>          eigenvalue.
    !>

    RCONDV

    !>          RCONDV is DOUBLE PRECISION array, dimension (N)
    !>          RCONDV(j) is the reciprocal condition number of the j-th
    !>          right eigenvector.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.   If SENSE = 'N' or 'E',
    !>          LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
    !>          LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
    !>          For good performance, LWORK must generally be larger.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (2*N-2)
    !>          If SENSE = 'N' or 'E', not referenced.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if INFO = i, the QR algorithm failed to compute all the
    !>                eigenvalues, and no eigenvectors or condition numbers
    !>                have been computed; elements 1:ILO-1 and i+1:N of WR
    !>                and WI contain eigenvalues which have converged.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 303 of file dgeevx.f.

  subroutine sgeevx (character balanc, character jobvl, character jobvr, character sense, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, real, dimension( * ) scale, real abnrm, real, dimension( * ) rconde, real, dimension( * ) rcondv, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

    !>
    !> SGEEVX computes for an N-by-N real nonsymmetric matrix A, the
    !> eigenvalues and, optionally, the left and/or right eigenvectors.
    !>
    !> Optionally also, it computes a balancing transformation to improve
    !> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
    !> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
    !> (RCONDE), and reciprocal condition numbers for the right
    !> eigenvectors (RCONDV).
    !>
    !> The right eigenvector v(j) of A satisfies
    !>                  A * v(j) = lambda(j) * v(j)
    !> where lambda(j) is its eigenvalue.
    !> The left eigenvector u(j) of A satisfies
    !>               u(j)**H * A = lambda(j) * u(j)**H
    !> where u(j)**H denotes the conjugate-transpose of u(j).
    !>
    !> The computed eigenvectors are normalized to have Euclidean norm
    !> equal to 1 and largest component real.
    !>
    !> Balancing a matrix means permuting the rows and columns to make it
    !> more nearly upper triangular, and applying a diagonal similarity
    !> transformation D * A * D**(-1), where D is a diagonal matrix, to
    !> make its rows and columns closer in norm and the condition numbers
    !> of its eigenvalues and eigenvectors smaller.  The computed
    !> reciprocal condition numbers correspond to the balanced matrix.
    !> Permuting rows and columns will not change the condition numbers
    !> (in exact arithmetic) but diagonal scaling will.  For further
    !> explanation of balancing, see section 4.10.2 of the LAPACK
    !> Users' Guide.
    !>

Parameters
    BALANC

    !>          BALANC is CHARACTER*1
    !>          Indicates how the input matrix should be diagonally scaled
    !>          and/or permuted to improve the conditioning of its
    !>          eigenvalues.
    !>          = 'N': Do not diagonally scale or permute;
    !>          = 'P': Perform permutations to make the matrix more nearly
    !>                 upper triangular. Do not diagonally scale;
    !>          = 'S': Diagonally scale the matrix, i.e. replace A by
    !>                 D*A*D**(-1), where D is a diagonal matrix chosen
    !>                 to make the rows and columns of A more equal in
    !>                 norm. Do not permute;
    !>          = 'B': Both diagonally scale and permute A.
    !>
    !>          Computed reciprocal condition numbers will be for the matrix
    !>          after balancing and/or permuting. Permuting does not change
    !>          condition numbers (in exact arithmetic), but balancing does.
    !>

    JOBVL

    !>          JOBVL is CHARACTER*1
    !>          = 'N': left eigenvectors of A are not computed;
    !>          = 'V': left eigenvectors of A are computed.
    !>          If SENSE = 'E' or 'B', JOBVL must = 'V'.
    !>

    JOBVR

    !>          JOBVR is CHARACTER*1
    !>          = 'N': right eigenvectors of A are not computed;
    !>          = 'V': right eigenvectors of A are computed.
    !>          If SENSE = 'E' or 'B', JOBVR must = 'V'.
    !>

    SENSE

    !>          SENSE is CHARACTER*1
    !>          Determines which reciprocal condition numbers are computed.
    !>          = 'N': None are computed;
    !>          = 'E': Computed for eigenvalues only;
    !>          = 'V': Computed for right eigenvectors only;
    !>          = 'B': Computed for eigenvalues and right eigenvectors.
    !>
    !>          If SENSE = 'E' or 'B', both left and right eigenvectors
    !>          must also be computed (JOBVL = 'V' and JOBVR = 'V').
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A. N >= 0.
    !>

    A

    !>          A is REAL array, dimension (LDA,N)
    !>          On entry, the N-by-N matrix A.
    !>          On exit, A has been overwritten.  If JOBVL = 'V' or
    !>          JOBVR = 'V', A contains the real Schur form of the balanced
    !>          version of the input matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    WR

    !>          WR is REAL array, dimension (N)
    !>

    WI

    !>          WI is REAL array, dimension (N)
    !>          WR and WI contain the real and imaginary parts,
    !>          respectively, of the computed eigenvalues.  Complex
    !>          conjugate pairs of eigenvalues will appear consecutively
    !>          with the eigenvalue having the positive imaginary part
    !>          first.
    !>

    VL

    !>          VL is REAL array, dimension (LDVL,N)
    !>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
    !>          after another in the columns of VL, in the same order
    !>          as their eigenvalues.
    !>          If JOBVL = 'N', VL is not referenced.
    !>          If the j-th eigenvalue is real, then u(j) = VL(:,j),
    !>          the j-th column of VL.
    !>          If the j-th and (j+1)-st eigenvalues form a complex
    !>          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
    !>          u(j+1) = VL(:,j) - i*VL(:,j+1).
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of the array VL.  LDVL >= 1; if
    !>          JOBVL = 'V', LDVL >= N.
    !>

    VR

    !>          VR is REAL array, dimension (LDVR,N)
    !>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
    !>          after another in the columns of VR, in the same order
    !>          as their eigenvalues.
    !>          If JOBVR = 'N', VR is not referenced.
    !>          If the j-th eigenvalue is real, then v(j) = VR(:,j),
    !>          the j-th column of VR.
    !>          If the j-th and (j+1)-st eigenvalues form a complex
    !>          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
    !>          v(j+1) = VR(:,j) - i*VR(:,j+1).
    !>

    LDVR

    !>          LDVR is INTEGER
    !>          The leading dimension of the array VR.  LDVR >= 1, and if
    !>          JOBVR = 'V', LDVR >= N.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>          ILO and IHI are integer values determined when A was
    !>          balanced.  The balanced A(i,j) = 0 if I > J and
    !>          J = 1,...,ILO-1 or I = IHI+1,...,N.
    !>

    SCALE

    !>          SCALE is REAL array, dimension (N)
    !>          Details of the permutations and scaling factors applied
    !>          when balancing A.  If P(j) is the index of the row and column
    !>          interchanged with row and column j, and D(j) is the scaling
    !>          factor applied to row and column j, then
    !>          SCALE(J) = P(J),    for J = 1,...,ILO-1
    !>                   = D(J),    for J = ILO,...,IHI
    !>                   = P(J)     for J = IHI+1,...,N.
    !>          The order in which the interchanges are made is N to IHI+1,
    !>          then 1 to ILO-1.
    !>

    ABNRM

    !>          ABNRM is REAL
    !>          The one-norm of the balanced matrix (the maximum
    !>          of the sum of absolute values of elements of any column).
    !>

    RCONDE

    !>          RCONDE is REAL array, dimension (N)
    !>          RCONDE(j) is the reciprocal condition number of the j-th
    !>          eigenvalue.
    !>

    RCONDV

    !>          RCONDV is REAL array, dimension (N)
    !>          RCONDV(j) is the reciprocal condition number of the j-th
    !>          right eigenvector.
    !>

    WORK

    !>          WORK is REAL array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.   If SENSE = 'N' or 'E',
    !>          LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
    !>          LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
    !>          For good performance, LWORK must generally be larger.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (2*N-2)
    !>          If SENSE = 'N' or 'E', not referenced.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if INFO = i, the QR algorithm failed to compute all the
    !>                eigenvalues, and no eigenvectors or condition numbers
    !>                have been computed; elements 1:ILO-1 and i+1:N of WR
    !>                and WI contain eigenvalues which have converged.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 303 of file sgeevx.f.

  subroutine zgeevx (character balanc, character jobvl, character jobvr, character sense, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) w, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, double precision, dimension( * ) scale, double precision abnrm, double precision, dimension( * ) rconde, double precision, dimension( * ) rcondv, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)

ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

    !>
    !> ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
    !> eigenvalues and, optionally, the left and/or right eigenvectors.
    !>
    !> Optionally also, it computes a balancing transformation to improve
    !> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
    !> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
    !> (RCONDE), and reciprocal condition numbers for the right
    !> eigenvectors (RCONDV).
    !>
    !> The right eigenvector v(j) of A satisfies
    !>                  A * v(j) = lambda(j) * v(j)
    !> where lambda(j) is its eigenvalue.
    !> The left eigenvector u(j) of A satisfies
    !>               u(j)**H * A = lambda(j) * u(j)**H
    !> where u(j)**H denotes the conjugate transpose of u(j).
    !>
    !> The computed eigenvectors are normalized to have Euclidean norm
    !> equal to 1 and largest component real.
    !>
    !> Balancing a matrix means permuting the rows and columns to make it
    !> more nearly upper triangular, and applying a diagonal similarity
    !> transformation D * A * D**(-1), where D is a diagonal matrix, to
    !> make its rows and columns closer in norm and the condition numbers
    !> of its eigenvalues and eigenvectors smaller.  The computed
    !> reciprocal condition numbers correspond to the balanced matrix.
    !> Permuting rows and columns will not change the condition numbers
    !> (in exact arithmetic) but diagonal scaling will.  For further
    !> explanation of balancing, see section 4.10.2 of the LAPACK
    !> Users' Guide.
    !>

Parameters
    BALANC

    !>          BALANC is CHARACTER*1
    !>          Indicates how the input matrix should be diagonally scaled
    !>          and/or permuted to improve the conditioning of its
    !>          eigenvalues.
    !>          = 'N': Do not diagonally scale or permute;
    !>          = 'P': Perform permutations to make the matrix more nearly
    !>                 upper triangular. Do not diagonally scale;
    !>          = 'S': Diagonally scale the matrix, ie. replace A by
    !>                 D*A*D**(-1), where D is a diagonal matrix chosen
    !>                 to make the rows and columns of A more equal in
    !>                 norm. Do not permute;
    !>          = 'B': Both diagonally scale and permute A.
    !>
    !>          Computed reciprocal condition numbers will be for the matrix
    !>          after balancing and/or permuting. Permuting does not change
    !>          condition numbers (in exact arithmetic), but balancing does.
    !>

    JOBVL

    !>          JOBVL is CHARACTER*1
    !>          = 'N': left eigenvectors of A are not computed;
    !>          = 'V': left eigenvectors of A are computed.
    !>          If SENSE = 'E' or 'B', JOBVL must = 'V'.
    !>

    JOBVR

    !>          JOBVR is CHARACTER*1
    !>          = 'N': right eigenvectors of A are not computed;
    !>          = 'V': right eigenvectors of A are computed.
    !>          If SENSE = 'E' or 'B', JOBVR must = 'V'.
    !>

    SENSE

    !>          SENSE is CHARACTER*1
    !>          Determines which reciprocal condition numbers are computed.
    !>          = 'N': None are computed;
    !>          = 'E': Computed for eigenvalues only;
    !>          = 'V': Computed for right eigenvectors only;
    !>          = 'B': Computed for eigenvalues and right eigenvectors.
    !>
    !>          If SENSE = 'E' or 'B', both left and right eigenvectors
    !>          must also be computed (JOBVL = 'V' and JOBVR = 'V').
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A. N >= 0.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>          On entry, the N-by-N matrix A.
    !>          On exit, A has been overwritten.  If JOBVL = 'V' or
    !>          JOBVR = 'V', A contains the Schur form of the balanced
    !>          version of the matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    W

    !>          W is COMPLEX*16 array, dimension (N)
    !>          W contains the computed eigenvalues.
    !>

    VL

    !>          VL is COMPLEX*16 array, dimension (LDVL,N)
    !>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
    !>          after another in the columns of VL, in the same order
    !>          as their eigenvalues.
    !>          If JOBVL = 'N', VL is not referenced.
    !>          u(j) = VL(:,j), the j-th column of VL.
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of the array VL.  LDVL >= 1; if
    !>          JOBVL = 'V', LDVL >= N.
    !>

    VR

    !>          VR is COMPLEX*16 array, dimension (LDVR,N)
    !>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
    !>          after another in the columns of VR, in the same order
    !>          as their eigenvalues.
    !>          If JOBVR = 'N', VR is not referenced.
    !>          v(j) = VR(:,j), the j-th column of VR.
    !>

    LDVR

    !>          LDVR is INTEGER
    !>          The leading dimension of the array VR.  LDVR >= 1; if
    !>          JOBVR = 'V', LDVR >= N.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>          ILO and IHI are integer values determined when A was
    !>          balanced.  The balanced A(i,j) = 0 if I > J and
    !>          J = 1,...,ILO-1 or I = IHI+1,...,N.
    !>

    SCALE

    !>          SCALE is DOUBLE PRECISION array, dimension (N)
    !>          Details of the permutations and scaling factors applied
    !>          when balancing A.  If P(j) is the index of the row and column
    !>          interchanged with row and column j, and D(j) is the scaling
    !>          factor applied to row and column j, then
    !>          SCALE(J) = P(J),    for J = 1,...,ILO-1
    !>                   = D(J),    for J = ILO,...,IHI
    !>                   = P(J)     for J = IHI+1,...,N.
    !>          The order in which the interchanges are made is N to IHI+1,
    !>          then 1 to ILO-1.
    !>

    ABNRM

    !>          ABNRM is DOUBLE PRECISION
    !>          The one-norm of the balanced matrix (the maximum
    !>          of the sum of absolute values of elements of any column).
    !>

    RCONDE

    !>          RCONDE is DOUBLE PRECISION array, dimension (N)
    !>          RCONDE(j) is the reciprocal condition number of the j-th
    !>          eigenvalue.
    !>

    RCONDV

    !>          RCONDV is DOUBLE PRECISION array, dimension (N)
    !>          RCONDV(j) is the reciprocal condition number of the j-th
    !>          right eigenvector.
    !>

    WORK

    !>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.  If SENSE = 'N' or 'E',
    !>          LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
    !>          LWORK >= N*N+2*N.
    !>          For good performance, LWORK must generally be larger.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (2*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if INFO = i, the QR algorithm failed to compute all the
    !>                eigenvalues, and no eigenvectors or condition numbers
    !>                have been computed; elements 1:ILO-1 and i+1:N of W
    !>                contain eigenvalues which have converged.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 285 of file zgeevx.f.

Author

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